Abstract

A shootout is a popular mechanism to identify a winner of a match between two teams. It consists of rounds in which each team gets, sequentially, an opportunity to score a point. It has been shown empirically that shooting first or shooting second in a round has an impact on the scoring probability. This raises a fairness question: is it possible to specify a sequence such that identical teams have equal chance of winning? We show that, for a sudden death, no repetitive sequence can be fair. In addition, we show that the so-called Prohuet–Thue–Morse sequence is not fair. There is, however, an algorithm that outputs a fair sequence whenever one exists. We also analyze the popular best-of-$k$ shootouts and show that no fair sequence exists in this situation. In addition, we find explicit expressions for the degree of unfairness in a best-of-$k$ shootout; this allows sports administrators to asses the effect of the length of the shootout on the degree of unfairness.

中文翻译：

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枪战公平性的数学分析

摘要

点球大战是一种流行的机制，用于确定两支球队之间的比赛获胜者。它由几轮组成，每个团队依次获得得分机会。经验表明，在一轮中先投篮或第二投篮对得分概率有影响。这就提出了一个公平问题：是否可以指定一个顺序，使相同的球队有平等的获胜机会？我们表明，对于突然死亡，没有重复的序列是公平的。此外，我们表明所谓的 Prohuet-Thue-Morse 序列是不公平的。然而，有一种算法可以在存在时输出一个公平的序列。我们还分析了流行的最好的 $k$ 枪战，并表明在这种情况下不存在公平的顺序。此外，我们在最好的 $k$ 枪战中找到了对不公平程度的明确表达；这允许体育管理员评估点球大战的长度对不公平程度的影响。